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Integrability of certain Hamiltonian systems in $2D$ variable curvature spaces

Wojciech Szumiński, Adel A. Elmandouh

Abstract

The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability. They are given in terms of arithmetic restrictions on values of the parameters describing the system. We apply the obtained results to some examples to illustrate that the applicability of the obtained result is easy and effective. Certain new integrable examples are given. The findings highlight the applicability of the differential Galois approach in studying the integrability of Hamiltonian systems in curved spaces, expanding our understanding of nonlinear dynamics and its potential applications.

Integrability of certain Hamiltonian systems in $2D$ variable curvature spaces

Abstract

The objective of this work is to examine the integrability of Hamiltonian systems in spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability. They are given in terms of arithmetic restrictions on values of the parameters describing the system. We apply the obtained results to some examples to illustrate that the applicability of the obtained result is easy and effective. Certain new integrable examples are given. The findings highlight the applicability of the differential Galois approach in studying the integrability of Hamiltonian systems in curved spaces, expanding our understanding of nonlinear dynamics and its potential applications.

Paper Structure

This paper contains 7 sections, 4 theorems, 41 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Variational equations and the integrability
  3. Applications of Theorem \ref{['TH_MR']}
  4. Example 1
  5. Example 2
  6. Example 3
  7. Conclusions

Key Result

Theorem 1

Let us assume a Hamiltonian system is Liouville integrable in a neighborhood of a phase curve $\Gamma$ corresponding to a certain particular solution. Then, the identity component $\mathcal{G}_{0}$ of the differential Galois group $\mathcal{G}$ associated with the variational equations along $\Gamma

Figures (1)

  • Figure 1: Poincaré sections of system \ref{['EXA1']} on surface $r=1$ with $p_r>0$. Top: $m=-1, n=-3$, at the level $E=-0.4$; Bottom: $m=-2, n=-6$, at the level $E=-0.4$

Theorems & Definitions (5)

  • Theorem 1: Morales-Ruiz and Ramis N3
  • Theorem 2: Kimura
  • Theorem 3: Main Theorem
  • Corollary 1
  • proof